Affinity

1. Affinities

These are invertible transformations of the plane onto itself, which, fixing a coordinate system (not necessarily orthogonal or having equal unit-lengths on the axes), are defined by :

the matrix having non zero determinant

Affinities are defined by prescribing their values {Q₁, Q₂, Q₃} at three points of the plane {P₁, P₂, P₃}, both tripples of points being in general position.

The set of all affinities of the plane is a group G, containing the rotations, reflexions, translations and similarities. The main properties of affinities are: [1] They preserve lines, and ratios on them.
[2] They preserve parallelity.
[3] They preserve congruence by translations (yellow triangles).
[4] They multiply areas with D.

3. Additional properties

Affinities transform conics to conics of the same kind. Besides they preserve conjugation of diameters. In particular they map orthogonal diameters of circles to conjugate diameters of ellipses. Call two curves affine equivalent if one is a transform of the other through an affinity.
[1] All ellipses are affine equivalent to the circle x²+y² = 1.
[2] All hyperbolas are affine equivalent to the hyperbola xy = 1.
[3] All parabolas are affine equivalent to the parabola y = x².

4. Classification modulo affine equivalence

[1] The classification of real algebraic curves of degree three (cubics), with respect to affine equivalence, has its origins in Newton and Pluecker and was completed by M. Nadjafikah (arXiv.math. DG/0507383 v1. 19 Jul 2005). The curves fall in seven classes.
[2] For higher degrees the classification has not yet been done.
[3] A particular problem in the framework of classification is the determination of the isotropy subgroups H_J, of G, for the various classes J of equivalent curves. H_J consists of all affinities leaving invariant some typical member of the class J.
[4] In the case of non-degenerate conics all H_J are isomorphic to the group of rotations and reflections {W_x}, leaving fixed a point x of the plane. This is directly seen for ellipses, since all of them are affinely equivalent to the circle. The other cases can be handled by projectivities carrying the given conic to the circle.

5. Affine equivalence of an ellipse to itself

In the case of an ellipse (c), the affinities preserving it are the well-known conjugations along some diameter, and the conjugations along tangents of homothetic ellipses.
The above figure illustrates an affinity of the second kind F(A --> B) leaving invariant an ellipse. The affinity is defined by chords AB tangent (at their middle-point) to an ellipse b' which is homothetic to a' with respect to its center. Both a', b' are images of concentric circles through another affinity G. F corresponds, through that affinity G (conjugation by G: F = G*R*G^-1), to the usual rotations (R) of the circle about its center.
A consequence of this is that the area of the two sectors of the ellipse, determined by AB remain constant as AB takes the various positions of the tangents to (b').
See the corresponding figure for parabolas in ParabolaSymmetries.html .

6. Problem

What kind of transformation results if insead of the homothetic ellipse above we take an arbitrary ellipse b' lying entirely inside the ellipse a' ?

The application resulting by such a correspondence is not a homography. The above image indicates why.
There it is defined the homography F mapping points {A,B,C} of the circle a' correspondingly to {A',B',C'} of the same circle so that lines {AA', BB',CC'} are all tangent to the inner circle b'. This homography associates to each point X on a' point X'=F(X) on a', such that XX' is tangent to a conic (the red ellipse).
This conic, in general, does not coincide with the inner circle b'. Even the contact points (white) of the three lines {AA',BB',CC'} with the conic do not coincide with the contact points (green) of the same lines with the inner circle b'.

References

Audin, Michele Geometry Berlin, Springer, 2002
Coxeter, H. S. M. Introduction to Geometry, 2nd ed.. New York, John Wiley, 1972, p. 199.
M. Nadjafikah Classification of cubics up to affine transformations (arXiv.math. DG/0507383 v1. 19 Jul 2005).